Objective of the research The objective of the thesis is to establish the strong law of large numbers forsequences of random variables and arrays of fuzzy random variables in convexcombi
Trang 11 Rationale
In recent decades, some results on limit theorems of law of large numbersfor a collection of random variables in metric space have been researched andestablished by several authors In 1992, Herer introduced a concept of math-ematical expectation of random variables in a separable and complete metricspace (X, d) of negative curvature Then, Herer derived strong law of large num-bers for sequences of independent and identically distributed integrable randomvariables In 1997, using Herer’s definition of the mathematical expectation ofrandom variables in X and the approximation method by a sequence of discreterandom variables, Fitte derived an ergodic theorem and strong law of largenumbers for sequences of integrable random variables Some limit theorems ofmartingale in a metric space established by Herer (in 1992, 1997) and Sturm (in2002) In 2006, Ter´an and Molchanov introduced the concept of convex com-bination space, which is a metric space endowed with a convex combinationoperation Then, Ter´an and Molchanov constructed the concept of mathemati-cal expectation of random variables with values in a convex combination spaceand derived strong law of large numbers for sequences of pairwise independentand identically distributed random variables Hence, the researching on limittheorems of the law of large numbers for random variables in metric space is
an up-to-date tendency of the probability theory
The limit theorems on the law of large numbers and complete convergencealso research for double arrays and triangular arrays of random variables Youcan find the basic results of this area in the monographic book of Klesov (in2014) Note that, when extending the theorems for sequences of random vari-ables to arrays, then the results and methods used for the sequences are notalways applicable to the arrays Hence, the results on the researching of limittheorems for double arrays and triangular arrays of random variables in metricspace are interesting and more meanings
When researching the strong law of large numbers, people often consider theindependence of random variables A research direction on limit theorems of
Trang 2the law of large numbers is to substitute independent conditions with weakerconditions such as pairwise independent, blockwise m-dependent, blockwise andpairwise m-dependent This is a research direction that deserves attention.Researching the limit theorems of fuzzy random variables is very important
in theory and practice In practice, the theory of fuzzy random variables hasbeen researched and applied to such areas: information technology, image pro-cessing, control engineering and some other areas In theory, many problems
in theory of fuzzy random variables relate to classical probability theory Somethe limit theorems in classical probability theory were extended to fuzzy ran-dom variables Especially, the law of large numbers for fuzzy random variableshas been studied by many researchers For example, Colubi (in 1999) derivedthe strong law of large numbers for independent and identically distributedfuzzy random variables in Rd Proske and Puri (in 2002) proved the stronglaw of large numbers for independent and identically distributed fuzzy randomvariables in Banach space Inoue (in 1991) obtained the law of large numbersfor sums of independent tight fuzzy random variables, which extends the re-sult of Taylor and Inoue (in 1985) for set-valued random variables Recently,Kim (in 2013) established weak law of large numbers for the weighted sum offuzzy random variables taking values in a real separable Banach space Hence,researching the limit theorems of the law of large numbers for fuzzy randomvariables in metric space is a research direction that has many meanings andvalues
With the above reasons, we have chosen the topic for the thesis that is:
“Some types of laws of large numbers for sequences and arrays ofrandom variables in convex combination space”
2 Objective of the research
The objective of the thesis is to establish the strong law of large numbers forsequences of random variables and arrays of fuzzy random variables in convexcombination space, to establish complete convergence and strong law of largenumbers for triangular arrays of random variables and triangular arrays of fuzzyrandom variables in convex combination space, to establish mean convergenceand weak law of large numbers for sequences of fuzzy random variables in convexcombination space under various assumptions
Trang 33 Subject of the research
- The strong law of large numbers for sequences of random variables, doublearrays of fuzzy random variables
- The complete convergence and strong law of large numbers for triangulararrays of random variables, triangular arrays of fuzzy random variables
- The mean convergence and weak law of large numbers for sequences offuzzy random variables
4 Scope of the research
Some types of convergence of sequences and arrays of random variables inconvex combination space The types of convergence are considered: almostsure convergence, complete convergence, mean convergence and convergence inprobability
5 Methodology of the research
We use the methods of probability and characteristics of analysis as: proximation method, truncation method, properties of a compact set, etc
ap-6 Contribution of the thesis
The results of thesis contribute more abundant for the researching directions
of general limit theorems and limit theorems for random variables in metricspace The thesis is contribution of material for the students, the master stu-dents, the doctoral students belonging to the speciality Theory of Probabilityand Mathematical Statistics
7 Organization of the research
7.1 Overview of the research
In this thesis, using the definition of convex combination space of Ter´an andMolchanov (in 2006), we establish limit theorems on the strong law of largenumbers, complete convergence, mean convergence and weak law of large num-bers for sequences and arrays of random variables and fuzzy random variables inconvex combination space First, we present some concepts and basic properties
of convex combination space Next, we present concepts of compactly uniformlyintegrable and compactly uniformly r-th integrable in Ces`aro sense for a col-lection of random variables in convex combination space, which are naturally
Trang 4extended from the corresponding concepts in Banach space to convex tion space We establish the strong law of large numbers for sequences of ran-dom variables in convex combination space satisfying the condition: blockwiseand pairwise m-dependent and compactly uniformly integrable in Ces`aro sense,
combina-or blockwise m-dependent and identically distributed We also establish thecomplete convergence and the strong law of large numbers for triangular arrays
of rowwise independent and compactly uniformly integrable random variables
in convex combination space For triangular arrays of fuzzy random variables
in convex combination space, we establish the complete convergence and thestrong law of large numbers for rowwise independent and (α, α+)-levelwise com-pactly uniformly integrable fuzzy random variables For double arrays of fuzzyrandom variables in convex combination space, we establish the strong law oflarge numbers for fuzzy random variables satisfying the condition: independentand (α, α+)-levelwise compactly uniformly integrable, or pairwise independentand identically distributed Finally, we establish necessary and sufficient condi-tions on mean convergence and weak law of large numbers for sequences of fuzzyrandom variables The convergence of fuzzy random variables in this thesis isconsidered by metric d∞
7.2 The organization of the research
Besides the sections of usual notations, preface, general conclusions and ommendations, list of the author’s articles related to the thesis and references,the thesis is organized into three chapters
rec-Chapter 1 presents some basic knowledge of convex combination space andrandom variable in convex combination space This chapter is organized asfollows: Section 1.1 presents concept of convex combination space, some exam-ples of convex combination space, basic properties of convex combination space.Section 1.2 presents random variable in convex combination space, defines themathematical expectation and integrability of a random variable in convex com-bination space, types of convergence: almost sure convergence, complete con-vergence, mean convergence and convergence in probability for sequences andarrays of random variables in convex combination space, concept of compactlyuniformly integrable and compactly uniformly r-th integrable in Ces`aro sensefor a collection of random variables in convex combination space Section 1.3
Trang 5presents concept and basic properties of fuzzy random variable in convex nation space The knowledge of Chapter 1 is used to establish the main results
combi-of the next chapters
Chapter 2 presents some limit theorems on the law of large numbers forsequences and triangular arrays of random variables in convex combinationspace Section 2.1 presents the strong law of large numbers for sequences ofblockwise and pairwise m-dependent random variables in convex combination.Section 2.2 presents the complete convergence and strong law of large numbersfor triangular arrays of random variables in convex combination space
Chapter 3 is used to research some limit theorems on the law of large numbersfor sequences, triangular arrays and double arrays of fuzzy random variables inconvex combination space Section 3.1 presents concepts of (α, α+)-levelwisecompactly uniformly integrable and (α, α+)-levelwise compactly uniformly r-thintegrable in Ces`aro sense for a collection of fuzzy random variables in con-vex combination space Section 3.2 presents the complete convergence and thestrong law of large numbers for triangular arrays of fuzzy random variables inconvex combination space Section 3.3 presents the strong law of large num-bers for double arrays of fuzzy random variables in convex combination space.Section 3.4 presents the mean convergence and weak law of large numbers forsequences of fuzzy random variables in convex combination space
Trang 6CHAPTER 1THE PREPARATION KNOWLEDGE
In this chapter, we present some concepts and basic properties of convexcombination pace, some examples of convex combination space, concepts andproperties of a random variable and a fuzzy random variable in convex combi-nation space We present the concept of compactly uniformly integrable andcompactly uniformly r-th integrable in Ces`aro sense for a collection of randomvariables in convex combination space
1.1 Convex combination space
In this thesis, if not added assume, we suppose that (Ω, A, P ) is a completeprobability space, (X, d) is a separable and complete metric space, BX is theBorel σ-algebra on X Let c(X) be the family of all nonempty compact subsets
of X
We denoted by N (resp N0) (resp R) the set of all positive integers (resp.nonnegative integers) (resp real numbers)
For two real numbers m and n, we denote max{m, n} (resp min{m, n}) by
m ∨ n (resp m ∧ n) For each a ∈ R+, the logarithm to the base 2 of a ∨ 1 will
be denoted by log+a
On the metric space (X, d), we define a convex combination operation: for all
n > 2, numbers λ1, , λn > 0 that satisfy Pni=1λi = 1 and all u1, , un ∈ X,this operation produces an element of X, which is denoted by [λi, ui]ni=1 or[λ1, u1; ; λn, un] Assume that [1, u] = u for every u ∈ Xand that the followingaxioms are satisfied
• Axiom 1: (Commutativity)
[λi, ui]ni=1 = [λσ(i), uσ(i)]ni=1 for every permutation σ of {1, , n};
• Axiom 2: (Associativity)
Trang 71.1.1 Definition The metric space (X, d) endowed with a convex combinationoperation is referred to as the convex combination space.
Note that, in the general case, Ku and u are not identical An element u ∈ X
is called convexely decomposable element if for all n > 2 and λ1, , λn > 0 with
Pn
i=1λi = 1, then
u = [λi, u]ni=1
A set A ⊂ X is called convex if [λi, ui]ni=1 ∈ A for all ui ∈ A and {λi : 1 6
i 6 n, n > 1} ⊂ (0; 1) satisfying Pni=1λi = 1 For A ⊂ X, we denote as coAthe convex hull of A, which is the smallest convex subset that contains A, andcoA is the closed convex hull of A
1.1.11 Theorem If (X, d) is a convex combination space, then the space c(X)with the convex combination
[λi, Ai]ni=1 = {[λi, ui]ni=1 : ui ∈ Ai, for all i}
and the Hausdorff metric dH
dH(A, B) = max
supa∈A
infb∈Bd(a, b), sup
b∈B
infa∈Ad(b, a)
is also a convex combination space, where the convexification operator Kc(X) isgiven by
Kc(X)A = coKXA = co{KXu : u ∈ A}
Since (X, d) is a separable and complete metric space, (c(X), dH) is also aseparable and complete metric space
Trang 81.2 Random variable in convex combination space
From now on, we assume that (X, d) is a convex combination space
1.2.1 Definition A mapping X : Ω → X is called A-measurable if for all
B ∈ BX, then X−1(B) ∈ A The mapping A-measurable X is also called an
X-valued random variable
When an X-valued random variable X takes finite values, it is called a simplerandom variable
For each X-valued random variable X, we denote σ(X) = {X−1(B) : B ∈
BX} Then, σ(X) is the smallest sub-σ-algebra of A with respect to which X
is measurable The distribution of X-valued random variable X is a probabilitymeasure PX on BX defined by
PX(B) = P X−1(B), B ∈ BX.1.2.2 Definition The collection of X-valued random variables {Xi : i ∈ I}
is said to be independent (resp pairwise independent ) if the collection of algebras {σ(Xi) : i ∈ I} is independent (resp pairwise independent), and issaid to be identically distributed if all PXi, i ∈ I, are identical
σ-1.2.3 Definition (a) The sequence ofX-valued random variables {Xn : n > 1}
is said to be almost sure convergence to the X-valued random variable X as
n → ∞, if there exists A ∈ A such that P (A) = 0 and for all ω ∈ Ω \ A, then
limn→∞d(Xn, X)(ω) = 0
We denote Xn → X a.s (or Xn −→ X) as n → ∞.a.s.
(b) The double array of X-valued random variables {Xmn : m > 1, n > 1} issaid to be almost sure convergence to the X-valued random variable X-valuedrandom variable X as m ∨ n → ∞, if there exists A ∈ A such that P (A) = 0and for all ω ∈ Ω \ A, then
limm∨n→∞d(Xmn, X)(ω) = 0
We denote Xmn → X a.s (or Xmn −→ X) as m ∨ n → ∞.a.s.
(c) The sequence of X-valued random variables {Xn : n > 1} is said to becomplete convergence to the X-valued random variable X as n → ∞, if for all
Trang 9limn→∞Edp(Xn, X) = 0.
We denote Xn −→ X as n → ∞.Lp
(e) The sequence of X-valued random variables {Xn : n > 1} is said to beconvergence in probability to the X-valued random variable X as n → ∞, if forall ε > 0, then
limn→∞P d(Xn, X) > ε = 0
We denote Xn −→ X as n → ∞.P
Let X be an X-valued random variable that takes a distinct value xi for eachnon-null set Ωi with i = 1, , n The expectation of X is defined by
EX = [P (Ωi), Kxi]ni=1 (1.2.1)Note that, if X, Y are simple random variables then d(EX, EY ) 6 Ed(X, Y )
We fix u0 ∈ K(X) (by Axiom 5, K(X) 6= ∅) and u0 will be considered asthe special element of X Since the metric space X is separable, there exists
a countable dense subset {un : n > 1} of X For each k > 1, we define themapping ϕk : X → X such that ϕk(x) = umk(x), where
mk(x) = minni ∈ {0, , k} : d(ui, x) = min
0 6 j 6 kd(uj, x)o.Hence, we obtain d(u0, ϕk(x)) 6 2d(u0, x)
1.2.5 Definition The X-valued random variable X is said to be integrable ifd(u0, X) is the integrable real-valued random variable
The space of all integrable X-valued random variables will be denoted by
L1X Then, for X ∈ L1X, the expectation of X is defined by
EX := lim
k→∞Eϕk(X)
Trang 10By the approximation method, we also prove that if X, Y ∈ L1X, then
d(EX, EY ) 6 Ed(X, Y )
The Borel σ-algebra on c(X) is generated by the collection of sets
KV := {A ∈ c(X) : A ∩ V 6= ∅},where V is an open subset in X, and denoted by Bc(X)
1.2.7 Definition A mapping X : Ω → c(X) is called c(X)-valued randomvariable if for all B ∈ Bc(X), then X−1(B) ∈ A
Note that (c(X), dH) ia a separable and complete convex combination space,therefore the concepts and properties of the expectation of integrable c(X)-valued random variables are similar as integrable X-valued random variables
We denote the expectation of an integrable c(X)-valued random variable X by
Trang 11A collection {Xi : i ∈ I} of real-valued random variables is said to beuniformly bounded by a real-valued random variable X, if for all i ∈ I and
t > 0, then
P (|Xi| > t) 6 P (|X|> t)
A collection {Xi : i ∈ I} of real-valued random variables is said to bestochastically dominated by a real-valued random variable X, if there exists aconstant C (0 < C < ∞) such that for all i ∈ I and t > 0, then
P (|Xn| > t) 6 CP (|X| > t)
1.3 Fuzzy random variable in convex combination space
A mapping v : X → [0; 1] is said to be fuzzy set on X We denote by F (X)the space of all fuzzy set v that satisfy: v is upper semicontinuous, sup v = 1and supp v = cl{x ∈ X : v(x) > 0} is compact in X For v ∈ F (X), its α-levelset Lαv is defined
d∞(v1, v2) = sup
α∈(0;1]
dH(Lαv1, Lαv2)
Trang 12is a convex combination space, where the convexification operator KF (X) isgiven by
dpH(Lαv1, Lαv2)dα
1/p, p > 1
is a convex combination space, where the convexification operator KF (X) isgiven by
Lα(KF (X)v) = Kc(X)Lαv, α ∈ (0; 1]
For v ∈ F (X), we denote kvk∞ = d∞(v, I{u0}) = supα>0kLαvk{u0}, where
I{u0} is the upper semicontinuous function that takes values 1 at u0 and 0 forall x 6= u0
1.3.3 Definition A mapping X : Ω → F (X) is said to be a fuzzy randomvariable in convex combination space X if LαX is a c(X)-valued random variablefor all α ∈ (0; 1]
From the above definition and (1.3.2) we have if X is a fuzzy random variable
in convex combination space X, then L+αX is a c(X)-valued random variable,for all α ∈ [0; 1)
A fuzzy random variable in convex combination space X is called integrablybounded if kL+0 Xk{u0} ∈ L1
R, then we denote X ∈ L1(F (X))
1.3.5 Definition The expectation of X ∈ L1(F (X)), denoted by EF (X)X, is
a fuzzy set on X such that for each α ∈ (0; 1], then
Lα(EF (X)X) = Ec(X)(LαX)
1.3.7 Definition A collection {Xi : i ∈ I} of fuzzy random variables in convexcombination space X is said to be independent (resp pairwise independent ) if{LαXi : i ∈ I} is a collection of independent (resp pairwise independent)c(X)-valued random variable, for each α ∈ (0; 1]
The conclusions of Chapter 1
In this chapter, we present concepts and results about convex combinationspace, random variable in convex combination space, fuzzy random variable in